Orders with low crossing number, introduced by Welzl, are a fundamental tool in range searching and computational geometry. Recently, they have found important applications in structural graph theory: set systems with linear shatter functions correspond to graph classes with linear neighborhood complexity. For such set systems S = (U, F), Welzl's theorem guarantees the existence of orders with only O(log^2 n) crossings, n = |U|. A series of works has progressively improved the runtime for computing such orders, from Chazelle and Welzl's original O(|U|^3 |F|) bound (Chazelle and Welzl, 1989), through Har-Peled's O(|U|^2|F|) (Har-Peled, 2009), to the recent sampling-based methods of Csikós and Mustafa (Csikós and Mustafa, 2024). We present a randomized algorithm that computes Welzl orders for set systems with linear primal and dual shatter functions in expected time O(||S|| log ||S||), where ||S|| = |U| + sum_{X in F} |X| is the size of the canonical input representation. The algorithm achieves an asymptotically optimal O(log n) crossing bound. As an application, we compute compact neighborhood covers in graph classes with (near-)linear neighborhood complexity in time O((n+m) log n), where n and m are the numbers of vertices and edges. Our result also improves the runtime of first-order model checking on monadically stable graph classes from O(n^(5+epsilon)) to O(n^(3+epsilon)).

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